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# Express the number as a ratio of intergers.$0. \overline 8 = 0.8888 . . .$

## $$\frac{8}{9}$$

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it's expressed this given number as a ratio of two imagers. What that means is we'd like to write this given number this decimal in the form of peel Rick. You. We're peeing Q or numbers such as, like negative, too negative wine zero one two and so on. Positive or negative? No fractions, But people like you will be our fraction. So here the way to do this is to write this as a geometric. So that's right as a geometric, Siri's. So the way to do that here it is, too. Let me raise this and you come down here some just going to rewrite this. So this is point eight and then point Oh, eight point zero zero eight and so on. And then here I can write. This first term is eight over ten plus and that here it looks like we move the decimal one place of the left. So we just multiplied by another ten on the bottom. Same thing over here, another place to the left. So that's ten cubed and we could see the pattern there. Now we can see its geometric and for a geometric series Not that this comm urges and our problems. Cube's Our is the number that were multiplying by each time. What are we most applying to go from here to here? It's one over ten. So in particular we have. This is true, so we know it emerges and we know the sum for geometric is the first term of the Siri's, all divided by one minus R. Now, in our problem, the first term is it over there and then we divide by one minus. R and R is one over ten. Let's go to simplify that, that's eight over ten overnight over ten and then cancel those tens and Europe with end up with eight over nine. And we've completed the task because we started with a decimal, and then we wrote it as a ratio of two imagers a day and night. That's our final answer

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